ALTERNATIVE TO THE HYPERCOMPLEX FOURIER TRANSFORM DEFINITION

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ISSN: 1311-1728 (printed version); ISSN: 1314-8060 (on-line version) doi:_{http://dx.doi.org/10.12732/ijam.v29i6.5}

C.A.P. Martinez1, A.L.M. Martinez2§ 1,2_DAMAT

Federal Technological University of Paraná CEP: 86300-000, Cornélio Procópio, PR, BRASIL

Abstract: In this paper we present a construction of the quaternionic hyper-complex version of Fourier Series and an introduction of a simple quaternionic expansion of Fourier Transform in hypercomplex exponentials is provided. By some examples, we show how the representation is written by using the pro-posed expansion.

AMS Subject Classification: 39G99, 30E99

Key Words: hypercomplex functions, quaternions, hypercomplex Fourier transform

1. Introduction and Motivation

The hypercomplex numbers are mostly concerned with quaternions and octo-nions. The involved algebras may be considered as extensions of the ordinary bi-dimensional complex algebra, according to [10], [2], the operation is non-commutative for quaternions.

The quaternions numbers were discovered in 1843 by Willian R. Hamilton (see [5]) and in that same year John T. Graves, a Hamilton’s friend, found an 8-dimensional algebra whose property on non-associativity in a multiplication table holds. Two years later, in 1845, after some contributions on this subject by Arthur Cayley, the octonions have also been named as “Cayley numbers”.

Received: August 29, 2016 c 2016 Academic Publications

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Among many possible methods for estimating complex-valued functions, Fourier Series are specifically attractive because the uniform convergence of the Fourier series (as more terms are added) is guaranteed for continuous and bounded functions. Furthermore, the Fourier coefficients are designed to min-imize the square of the error from the actual function. Finally, it is relatively simple to deal with complex exponentials and these frequently occur in physical phenomena.

The Fourier Transform is very important also in theoretical mathematics area. The Fourier Transform of time function is already a complex-valued func-tion of frequency, whose absolute value represents the amount of that frequency in the original function, and whose complex argument is the phase offset of the basic sinusoid in that frequency.

In this paper, based on the previous work (Pendeza et al., see in [8]), a simple quaternionic expansion of Fourier transform in hypercomplex exponentials is proposed. Through some examples, we show how the representation is written by using the proposed expansion, in order to approach practical problems in a near future.

2. Hypercomplex Fourier Series

The quaternions form the 4-dimensional normed division algebra on R. The quaternions algebra denoted by Q, is an alternative division algebra. The quaternions setQ is defined by Q={a, b, c, d} ∈ R,where

(a, b, c, d) = (a′, b′, c′, d′)⇐⇒ a=a′, b=b′, c=c′, d=d′.

The quaternions do not form a ring due to the non-commutativity of the multi-plication. Also they do not form a group due to a nonassociativide of multipli-cation. They form aMoufang Loop, a Loop with identity element (N. Jacobson, [11]).

Let us consider a quaternionic number given by

q =a+bi+cj+dk.

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a0 2 +

∞

X

n=1 h

ancos

_nπx L

+bnsin

_nπx L

i

, (1)

is convergent and the limit is fe(x) = lima→x+f(x) + lima→x−f(x)

2 .

The coefficients of Fourier off,a0,anand bn are given by:

a0 = 1 L

Z L

−L

f(x)dx,

an=

1 L

Z L

−L

f(x) cosnπx L

dx, (2)

and

bn=

1 L

Z L

−L

f(x) sinnπx L

dx. (3)

The trigonometric series presented in (1) with this choice of coefficients is the Fourier series off.

Now, we will detail some properties considering quaternion q∈ Qgiven by q =u1+u2i+u3j+u4k=u1+−→u .

The equation of De Moivre for quaternions is given by

eq=eu1

( cos

q

u2₂+u2₃+u2₄ !

+~u sin( p

u2₂+u2₃+u2₄) p

u2

2+u23+u24 !)

. (4)

Then, using (4) we can obtain:

eiy+e−iy

2 =

e0(cos|y|+iysin_|y||y|) +e0(cos| −y| −iysin_|−y||−y|)

2 .

Since cos| −y|= cos|y|= cos(y) and sin| −y|) = sin|y|, we have eiy₊_e−iy

2 = cos(y).

Following the same arguments, we can obtain

cosy= e

iy₊_e−iy

2 =

ejy₊_e−jy

2 =

eky₊_e−ky

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and

siny= e

iy₋_e−iy

2i =

ejy₋_e−jy

2j =

eky₋_e−ky

2k . (6)

From (5) and (6), we write

cosy= (e

iy₊_ejy₊_eky_{) + (e}−iy₊_e−jy₊_e−ky₎

6 ,

siny= 1 3

eiy₋_e−iy

2i +

ejy₋_e−jy

2j +

eky₋_e−ky

2k

. Consequently,

ancos

_nπx L

+bnsin

_nπx L

=an

"

(ei(nπxL )+ej( nπx

L )+ek( nπx

L ))

6

+ (e

−i(nπx_L )₊_e−j(nπx_L )₊_e−k(nπx_L )₎ 6

# +bn

3 "

ei(nπxL )−e−i( nπx

L )

2i

+e

j(nπx

L )−e−j( nπx

L )

2j +

ek(nπxL )−e−k( nπx L ) 2k # = 1 3       an 2 + bn 2i

| {z }

c1 n

einπxL +

an 2 − bn 2i

| {z }

c1

−n

e−inπxL +

an 2 + bn 2j

| {z }

c2 n

ejnπxL

+ an 2 − bn 2j

| {z }

c2 −n

e−jnπxL +

an 2 + bn 2k

| {z }

c3 n

eknπxL +

an 2 − bn 2k

| {z }

c3 −n

e−(k)nπxL

      .

By considering that

c1_n = an 2 +

bn

2i = 1

2(an−ibn), and using (2) and (3), we have

c1_n= 1 2L

Z L

−L

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Again, using the same argument we can obtain

c1_−n= 1 2L

Z L

−L

f(x)e−i ₍

−n)πx

L

dx. (8)

Extending the used ideas, we conclude that

c2_n= 1 2L

Z L

−L

f(x)e−j(nπxL )dx, (9)

since (j)2 =−1,

c3_n= 1 2L

Z L

−L

f(x)e−k(nπxL )dx, (10)

since (k)2=−1. From (7), (8), (9) and (10) we can write

∞

X

n=1 h

ancos

_nπx L

+bnsin

_nπx L i = 1 3 ∞ X −∞,n6=0 h

c1_neinπxL +c2

ne

jnπx L +c3

ne

knπx L

i .

Sincef :R→R is periodic, with period 2L,f andf′ are piecewise continuous, we have that the Fourier series of f presented in (1) can be written as follows:

e

f(x) =co+

1 3    ∞ X −∞,n6=0

c1_neinπxL +c2

ne

jnπx L +c3

ne

knπx L



, (11)

by consideringc0 = a₂0.

The series (11) is the Hypercomplex Fourier Series of f.

3. The Hypercomplex Fourier Transform

In this section we will consider the definition of Hypercomplex Fourier Series (11) to deduce a model for Hypercomplex Fourier Transform. From equation (11), consider f defined in the interval (−L.L) with f and f′ are piecewise continuous. Denoting αn= nπ_L, we obtain

e

f(x) = 1 2L

Z L

−L

f(u)du+ 1 6L ∞ X −∞,n6=0 Z L −L

f(u)e−i(αnu)_du

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+ Z L

−L

f(u)e−j(αnu)_du

ejαnx₊

Z L

−L

f(u)e−k(αnu)_du

ekαnx

, considering ∆α=αn+1−αn,

= ∆α 2π

Z L

−L

f(u)du+ 1 6π ∞ X −∞,n6=0 Z L −L

f(u)e−i(αnu)_du

eiαnx

+ Z L

−L

f(u)e−j(αnu)_du

ejαnx₊

Z L

−L

f(u)e−k(αnu)_du

ekαnx

∆α,

calculating the limit ∆α→0 and assumingR_−∞∞ |f(u)|du <∞

= lim ∆α→0

1 6π ∞ X −∞,n6=0 Z L −L

f(u)e−i(αnu)_du

eiαnx

+ Z L

−L

f(u)e−j(αnu)_du

ejαnx₊

Z L

−L

f(u)e−k(αnu)_du

ekαnx

∆α.

From the definition of the Riemann integral, we obtain

= 1 6π Z ∞ −∞ Z ∞ −∞

f(u)e−i(αu)du

eiαx+

Z ∞

−∞

f(u)e−j(αu)du

ejαx

+ +

Z ∞

−∞

f(u)e−k(αu)du

ekαx

dα. Or, equivalently, we have

1 6π Z ∞ −∞ Z ∞ −∞

f(u)e−i(αu)du, Z ∞

−∞

f(u)e−j(αu)du, Z ∞

−∞

f(u)e−k(αu)du,

(eiαx, ejαx, ekαx)

dα, (12) denoting

Fi(α) =

Z ∞

−∞

f(u)e−i(αu)du, Fj(α) =

Z ∞

−∞

f(u)e−j(αu)du,

Fk(α) =

Z ∞

−∞

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Thus (12) can be written as

e

f(x) = 1 6π

Z ∞

−∞

D

(Fi(α), Fj(α), Fk(α)),(eiαx, ejαx, ekαx)

E

dα. (13) We define transformed as Hypercomplex,

F_H(α) =T_H_(f_{(x)) = (F}_i_{(α), F}_j_{(α), F}_k_(α)).

The inverse transform is calculated by the equation

T−1

H =

1 6π

Z ∞

−∞

D

F_H(α),(eiαx, ejαx, ekαx)Edα,

We present below a numerical example for the Hypercomplex Fourier Transform calculation.

Example 1. Consider the function f(x) =

1, for −π < x <0

−1, for 0< x < π . Computing the Hypercomplex Fourier Transform we obtain

F_H(α) =T_H_(f_(x))

= 2

icos(απ)−1 α , j

cos(απ)−1 α , k

cos(απ)−1 α

.

Example 2. The Dirac delta (see [8]) can be poorly thought as a function on the real line which is zero everywhere except at the origin, where it is infinite,

δ(x) = (

+∞, x= 0 0, x6= 0 and which is also constrained to satisfy the identity

Z ∞

−∞

δ(x)dx= 1.

The delta function is a tempered distribution and, therefore, it has a well-defined Fourier transform.

T_{(δ(ξ)) =}

Z ∞

−∞

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Thus, calculating

F_H(α) = T_H_(δ(ξ))

= (Fi(α), Fj(α), Fk(α))

= (1,1,1),

we obtain consequently,

T−1

H (1,1,1) =

1 6π

Z ∞

−∞

D

(1,1,1),(eiαx_{, e}jαx_{, e}kαx₎E_dα

= δ(ξ).

4. Concluding Remarks

In order to generalize the Fourier transform to its quaternionic form, first we described Fourier series and we introduced the hypercomplex model by consid-ering results from [8, 9], which unable to obtain a formulation for hypercomplex Fourier transform, which depends on a product internal. Furthermore, since not few models of Theoretical Physics may be analyzed through the geometry and algebra of hypercomplex, it will be our concern to concentrate the next steps in making all possible applications of our results in the context of unified physical theories for higher dimensional space-times.

References

[1] L. Sinegre, Quaternions and the motion of a solid body about a fixed point according to Hamilton,Rev.-Historie-Math., No 1 (1995), 83-109.

[2] T.Y. Lam,Handbook of Algebra, Vol. 3, North-Holland, Amsterdam (2003), 429-454.

[3] S. Eilenberg, I. Niven, The fundamental theorem of algebra for quater-nioins, Bull. Amer. Math. Society,50 (1944), 246-248.

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[5] J. Mar˜ao, M.F. Borges, Liouville’s theorem and power series for quater-nionic functions,International Journal of Pure and Appllied Mathematics, 71, No 3 (2011), 383-384.

[6] S. Bock, K. Gurlebeck, On a generalized Appell system and monogenic power series,Math. Methds Appl. Sci.,33, No 4 (2010), 395-411.

[7] W. Rudin, Principles of Mathematical Analysis, 3rd Ed., MacGraw-Hill (1976).

[8] C.A.P. Martinez, M.F. Borges, A.L.M. Martinez and E.V. Castelani, Fourier series for quaternions and the extension of the square of the er-ror theorem, International Journal of Applied Mathematics, 25 (2012), 559-570.

[9] C.A.P. Martinez, M.F. Borges, A.L.M. Martinez and E.V. Castelani, Square of the error octonionic theorem and hypercomplex Fourier series, Tendˆencias em Matem´atica Aplicada e Computacional,14(2013), 483-495. [10] K. Kodairo, Complex Analysis, Cambridge University Press, Cambridge,

2007.

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